Quantum Cosmology with Distorted Gravity

transcript

Quantum Cosmologywith

Distorted Gravity

Remo GarattiniUniversità di Bergamo

I.N.F.N. - Sezione di Milano

Cagliari 18 Marzo 2016

Plan of the Talk

•Building the Wheeler-DeWitt Equation•The Wheeler-DeWitt Equation as a Sturm-Liouville problem•Relaxing the Lorentz Symmetry in a MSS approach for a FLRW model.•The Cosmological Constant as a Zero Point EnergyComputation in the Gravity’s Rainbow context

•Conclusions and Outlooks

34 4 31 2 2 82 Newton's Constant Cosmological Constant

matterS d x g R d x g K S G

Relevant Action for Quantum Cosmology

34 4 31 122 matterS d x g R d x g K S

Relevant Action for Quantum Cosmology

i i jiji ij

NN N N N N Ng g

N g N N NgN N

ADM Decomposition

2 2 2ds

is the lapse function is the shift function

i i j jij

g dx dx N dt g N dt dx N dt dx

ijij ij i j j i ijK g N N K K g

33 2 3

Legendre Transformation

2 2 0 Classical Constraint Invariance by time2

ijij matterI

ij klijkl

S dtd xN g K K K R S S

H d x N N H

reparametrization2 0 Classical Constraint Gauss Lawi ij

2 2 22

i i jiji ij

NN N N N N Ng g g dx dx

N g N N NgN N

Wheeler-De Witt EquationB. S. DeWitt, Phys. Rev.160, 1113 (1967).

2 2 02

ij klijkl ij

gG R g

• Gijkl is the super-metric,• R is the scalar curvature in 3-dim.

2 2 2 2 23ds N dt a t d

2 42 2

9 04 3

qH a a a aa a a G

Formal Schrödinger Equation with zero eigenvalue whose solution is a linear combination ofAiry’s functions (q=-1 Vilenkin Phys. Rev. D 37, 888 (1988).) containing expanding solutions

Example:WDW for Tunneling

E=0 is highly degenerate

Sturm-Liouville Eigenvalue Problem

1 9 04 3

qqH a a a a a E a E

a a a G

* Normalization with weight b

d dp x q x w x y xdx dx

w x y x y x dx w x

2 43 3 2 2 3

q q qp x a t q x a t w x a t y x aG G

qa a a da

Sturm-Liouville Eigenvalue Problem Variational procedure

1 9 04 3

qqH a a a a a

a a a G

d dy x p x q x y x dxdx dx

w x y x y x dx

0y a y b

Rayleigh-RitzVariational Procedure

Sturm-Liouville Eigenvalue Problem Variational procedure

1 9 04 3

qqH a a a a a

a a a G

3*23 min

d da a a a dada da G

Ga a a da

0 0 0 De Witt Condition

Rayleigh-RitzVariational Procedure

0 0 for example for q=0

exp No Solutiona a

Relaxing Lorentz symmetryHořava-Lifshitz theory UV Completion, problems with scalar graviton in IR

Varying Speed of Light Cosmology Solve problems in the Inflationary phase(horizon,flatness, particle production)

Gravity’s Rainbow Like VSL. Moreover it allows finite calculation to one loop.The set of the Rainbow’s functions is too large. A selection procedure is necessary

At low energy all these models describe GR

Gravity’s Rainbow

2 2 2 2 21 2

1 2/ 0 / 0

lim / lim / 1P P

P PE E E E

E g E E p g E E m

g E E g E E

Doubly Special Relativity

G. Amelino-Camelia, Int.J.Mod.Phys. D 11, 35 (2002); gr-qc/001205.G. Amelino-Camelia, Phys.Lett. B 510, 255 (2001); hep-th/0012238.

Curved Space Proposal Gravity’s Rainbow[J. Magueijo and L. Smolin, Class. Quant. Grav. 21, 1725 (2004) arXiv:gr-qc/0305055].

2 2 2 22 2 2 2

2 2 21 2 22

sin/ / /

exp 2 is the redshift function

is the shape function Condition ,

P P PP

N r dt dr r rds d db rg E E g E E g E E

g E Er

N r r r

b r b r r r r

Gravity’s Rainbow Application to Inflation[R. Garattini and M. Sakellariadou, Phys. Rev. D 90 (2014) 4, 043521; arXiv:1212.4987 [gr-qc]]

22 2 2

2 22 2

2 22 22

3 /1 0

2 / 3 /

3 41 0 12 3

effeff

g E Eq aa aa a a Gg E E g E E

aq Ga a Va a a G

2 22 2 2

32 21 2

Distorted / /

N t a tds dt d FLRW metric

g E Ep g E Ep

22 2 22 2

2 21 2

3 /1 0

2 / 3 /P

g E Eq aa aa a a Gg E E g E E

But we can go beyond this…indeed if thenE E a t

2 2 2 21 2

1 33 / , where , 1 2ijij P

aK K K g E E f a t a f a t a a t A t A t a tN t a

Gravity’s Rainbow Application to Hořava-Lifshitz theory[R. Garattini and E.N.Saridakis, Eur.Phys.J. C75 (2015) 7, 343; arXiv:1411.7257 [gr-qc]]

dg E a t E dEA tdE dag E a t E E

2 2 2 2

2 22 2 2then using the "normal" dispersion relation and P

k k k kE Ea t a l G

2 422 1 22 4

If we fix / , =1

g E E f a t a

E a t E a tg E E c c

2 2 2 2

22 2 4 2

16 256 16 256/ 1 1 Pb G G b R Rg E E

a t a t R R

Potential part of the Projectable Hořava-Lifshitz theorywithout detailed balancedCondition z=3

It is possible to build a mapAlso for SSM

10 12 1 g g 0b c GR

Applying the Rayleigh-Ritz procedure we can find candidate eigenvaluesdepending on the combination of the coupling constants[R. G., P.R.D 86 123507 (2012) 7, 343; arXiv:0912.0136 [gr-qc]]

The WDW equation becomes

Albrecht, Barrow, Harko, Maguejio, Moffat..

MSS in a VSL CosmologyR.G. and M.De Laurentis, arXiv:1503.03677

0 PSetting a kl

A Brief Mention to GUP

[R. Garattini and Mir Faizal; N.P. B 905 (2016) 313 arXiv:1510.04423 [gr-qc]]

Flat space

Deformed Momentum

Deformed U.P.

Trial Wave Function

Higher Order Derivative

From Mini-SuperSpaceto

Field Theory in 3+1 Dimensions

The Cosmological Constantas a Zero Point Energy

Calculation

Generalization

Solve this infinite dimensional PDE with a VariationalApproach without matter fields contribution

is a trial wave functional of the gaussian typeSchrödinger Picture

Spectrum of depending on the metricEnergy (Density) Levels

D h h d x h

V D h h h

D h D h D D h J

InducedCosmological‘‘Constant’’

Eliminating Divergences usingGravity’s Rainbow

[R.G. and G.Mandanici, Phys. Rev. D 83, 084021 (2011), arXiv:1102.3803 [gr-qc]]

1,2 1323

1ˆ 2 , ,4 2

ijkl ijkl

g E g Ed x gG K x x K x x

V g E g E

, : (Propagator)2

h x h yK x y

Modified Lichnerowicz operator

Standard Lichnerowicz operator

aia j ijij

jl a aij ijkl ia j ja iij

h R h Rh

h h R h R h R h

E= ijijh h

One loop Graviton Contribution

We can define an r-dependent radial wave number

l lEk r l E m r r r xg E E r

21 2 2 3

22 2 2 3

3 ' 36 12 2

' 36 12 2

b r b r b rm r

r r r r

b r b r b rm r

r r r r

1 22 21 2*

1 ( / ) ( / ) ( )3 ( / )8

ii P P i i

i i PE

EdE g E E g E E m r dEdE g EG E

122 2 2

18 16 i

dG m r

Standard Regularization

Eliminating Divergences usingGravity’s Rainbow

[R.G. and G.Mandanici, Phys. Rev. D 83, 084021 (2011), arXiv:1102.3803 [gr-qc]]

Gravity’s Rainbow and the Cosmological ConstantR.G. and G.Mandanici, Phys. Rev. D 83, 084021 (2011), arXiv:1102.3803 [gr-qc]

Popular Choice...... Not Promising

Eg E EE

Failure of Convergence

/ exp 1

E Eg E EE E

2 2 2 2 21 2 0 0

Minkowski de Sitter Anti de Sitter

m r m r m r x m r E

Conclusions and Outlooks• The Wheeler De Witt equation can be considered as a

Sturm-Liouville Problem Rayleigh-Ritz Variationalprocedure.

• In ordinary GR, we need a cut-off or a regularization/renormalization scheme.

• Application of Gravity’s Rainbow can be considered tocompute divergent quantum observables.

• Neither Standard Regularization nor Renormalization are required. This also happens in NonCommutativegeometries. A tool for ZPE Computation

• A connection between Horava-Lifshits theory withoutdetailed balanced condition and with projectability and Gravity’s Rainbow seems possible, at least in a FLRW metric. This is expected also for a VSL

• Repeating the above procedure for a SSM• Technical Problems with Kerr and other complicated

metrics. Comparison with Observation.

Quantum Cosmology with Distorted Gravity

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